Analysis is one of the essential basic subjects of mathematics. Originally developed to be able to formulate and solve problems of classical mechanics mathematically, in its present rigorous form it has become indispensable in numerous applications in the natural sciences and technology. This module aims to introduce the basic tool of differential and integral calculus and to explain their mutual interrelations. In addition, the concept of differential equations is explained and corresponding solution methods are discussed for simple cases.

1. Sequences and Series
   1. Sequences: Convergence and Monotony
   2. Definition and Convergence
   3. Specific Sequences and Series
2. Functions and Inverse Functions
   1. Functions and Their Properties
   2. Exponential and Logarithmic Functions
   3. Trigonometric Functions
3. Differential Calculus
   1. First Derivative and Power Rule
   2. Differentiation Rules and Higher Derivatives
   3. Taylor Series and Taylor Polynomial
   4. Curve Sketching
   5. Outlook: Partial Derivatives
4. Integral Calculus
   1. The Indefinite Integral and Integration Rules
   2. The Definite Integral and the Fundamental Theorem of Calculus
   3. Volume and Lateral Surface of Solids of Revolution and Arc Length
5. Differential Equations
   1. Introduction and Basic Terms
   2. Solution of First-Order Linear Homogeneous Differential Equations
   3. Solution of First-Order Linear Non-Homogeneous Differential Equations
   4. Outlook: Partial Differential Equations

Linear algebra is a fundamental subject in mathematics. Its historical origin lies in the development of solution techniques for systems of linear equations arising from geometric problems. Numerous scientific and engineering applications can be solved using its methods.

This course introduces the foundations of linear algebra and its basic notions like vectors and matrices. It then builds upon this foundation by introducing the derivation of solution techniques for problems in analytical geometry.

1. Foundations
   1. Systems of Linear Equations
   2. Matrices: Basic Terms
   3. Matrix algebra
   4. Matrices as compact representations of linear equations
   5. Inverse and trace
2. Vector Spaces
   1. Definition
   2. Linear Combination and Linear Dependance
   3. Basis, Linear Envelope, and Rank
3. Linear and Affine Mapping
   1. Matrix Representation of Linear Mappings
   2. Image and Kernel
   3. Affine Spaces and Subspaces
   4. Affine Mapping
4. Analytical Geometry
   1. Norm
   2. Scalar Product
   3. Orthogonal Projections
   4. Outlook: Complex Numbers
5. Matrix Decomposition
   1. Determinant
   2. Eigenvalues and Eigenvectors
   3. Cholesky Decomposition
   4. Eigenvalue Decomposition and Diagonalisation
   5. Singular Value Decomposition

Statistical description and analysis are the foundations for data-driven analysis and prediction methods. This course introduces the fundamentals, beginning with a formal definition of probabilities and introduction to the concepts underlying Bayesian statistics.

Random variables and probability density distributions are then discussed, as well as the concept of joint and marginal distributions. The importance of various discrete and continuous distributions and their applications is stressed.

Characterizing distributions is an important aspect of describing the behavior of probability distributions. Students are familiarized with expectation values, variance, and covariance. The concepts of algebraic and central moments and moment-generating functions complement the characterization of probability distributions.

Finally, this course focuses on important inequalities and limit theorems such as the law of large numbers or the central limit theorem.

1. Probability
   1. Definitions
   2. Independent events
   3. Conditional probability
   4. Bayesian statistics
2. Random Variables
   1. Random Variables
   2. Distribution functions and probability mass functions
   3. Important discrete probability distributions
   4. Important continous probability distributions
3. Joint Distributions
   1. Joint distributions
   2. Marginal distributions
   3. Independent random variables
   4. Conditional distributions
4. Expectation and Variance
   1. Expectation of a random variable, conditional expectations
   2. Variance and covariance
   3. Expectations and variances of important probability distributions
   4. Algebraic and central moments
   5. Moment-generating functions
5. Inequalities and Limit Theorems
   1. Probability inequalities
   2. Inequalities for expectations
   3. The law of large numbers
   4. Central limit theorem

Statistical analysis and understanding are the foundations of data-driven methods and machine learning approaches.

This course gives a thorough introduction to point estimators and discusses various techniques to estimate and optimize parameters. Special focus is given to a detailed discussion of both statistical and systematic uncertainties as well as propagation of uncertainties.

Bayesian statistics is fundamental to data-driven approaches, and this course takes a close look at Bayesian techniques such as Bayesian parameter estimation and prior probability functions.

Furthermore, this course gives an in-depth overview of statistical testing and decision theory, focusing on aspects such as A/B testing, hypothesis testing, p-values, and multiple testing which are fundamental to statistical analysis approaches in a broad range of practical applications.

1. Point Estimation
   1. Method of moments
   2. Sufficient statistics
   3. Maximum likelihood
   4. Ordinary least squares
   5. Resampling techniques
2. Uncertainties
   1. Statistical and systematic uncertainties
   2. Propagation of uncertainties
3. Bayesian Inference & Non-parametric Techniques
   1. Bayesian parameter estimation
   2. Prior probability functions
   3. Parzen windows
   4. K-nearest-neighbours
4. Statistical Testing
   1. A/B testing
   2. Hypothesis tests & test statistics
   3. P-values & confidence intervals
   4. Multiple testing
5. Statistical Decision Theory
   1. The risk function
   2. Maximum likelihood, Minimax, and Bayes
   3. Admissibility and Stein's paradox